Is $f(x)=0$ Even or Odd function or both?

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At present I know that, a constant function is an EVEN function.

But my teacher said that if a constant function has value $0$ then it is both EVEN and ODD.

I just want to verify this and want know the reason for this.

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It's even because $0 = f(x) = f(-x) = 0$, and it's odd because $0 = -0 = -f(x) = f(-x) = 0$. So it's both!